Optimal. Leaf size=87 \[ \frac {i \sqrt {\pi } e^{-i a} \text {erfi}\left (\sqrt {1-i c} x\right )}{4 \sqrt {1-i c}}-\frac {i \sqrt {\pi } e^{i a} \text {erfi}\left (\sqrt {1+i c} x\right )}{4 \sqrt {1+i c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4472, 2204} \[ \frac {i \sqrt {\pi } e^{-i a} \text {Erfi}\left (\sqrt {1-i c} x\right )}{4 \sqrt {1-i c}}-\frac {i \sqrt {\pi } e^{i a} \text {Erfi}\left (\sqrt {1+i c} x\right )}{4 \sqrt {1+i c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rule 4472
Rubi steps
\begin {align*} \int e^{x^2} \sin \left (a+c x^2\right ) \, dx &=\int \left (\frac {1}{2} i e^{-i a+(1-i c) x^2}-\frac {1}{2} i e^{i a+(1+i c) x^2}\right ) \, dx\\ &=\frac {1}{2} i \int e^{-i a+(1-i c) x^2} \, dx-\frac {1}{2} i \int e^{i a+(1+i c) x^2} \, dx\\ &=\frac {i e^{-i a} \sqrt {\pi } \text {erfi}\left (\sqrt {1-i c} x\right )}{4 \sqrt {1-i c}}-\frac {i e^{i a} \sqrt {\pi } \text {erfi}\left (\sqrt {1+i c} x\right )}{4 \sqrt {1+i c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.24, size = 129, normalized size = 1.48 \[ -\frac {\sqrt [4]{-1} \sqrt {\pi } \left (\sqrt {c+i} \left (\sin (a) \text {erf}\left (\frac {(1+i) \sqrt {c+i} x}{\sqrt {2}}\right )+\text {erfi}\left ((-1)^{3/4} \sqrt {c+i} x\right ) (c \sin (a)+i c \cos (a)+\cos (a))\right )+\sqrt {c-i} (c+i) (\cos (a)+i \sin (a)) \text {erfi}\left (\sqrt [4]{-1} \sqrt {c-i} x\right )\right )}{4 \left (c^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 66, normalized size = 0.76 \[ \frac {\sqrt {\pi } {\left (c + i\right )} \sqrt {-i \, c - 1} \operatorname {erf}\left (\sqrt {-i \, c - 1} x\right ) e^{\left (i \, a\right )} + \sqrt {\pi } {\left (c - i\right )} \sqrt {i \, c - 1} \operatorname {erf}\left (\sqrt {i \, c - 1} x\right ) e^{\left (-i \, a\right )}}{4 \, {\left (c^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (x^{2}\right )} \sin \left (c x^{2} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 62, normalized size = 0.71 \[ -\frac {i \sqrt {\pi }\, {\mathrm e}^{i a} \erf \left (\sqrt {-i c -1}\, x \right )}{4 \sqrt {-i c -1}}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-i a} \erf \left (\sqrt {i c -1}\, x \right )}{4 \sqrt {i c -1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 137, normalized size = 1.57 \[ \frac {\sqrt {\pi } \sqrt {2 \, c^{2} + 2} {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, c - 1} x\right ) + {\left (\cos \relax (a) + i \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, c - 1} x\right )\right )} \sqrt {\sqrt {c^{2} + 1} + 1} - \sqrt {\pi } \sqrt {2 \, c^{2} + 2} {\left ({\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, c - 1} x\right ) + {\left (i \, \cos \relax (a) - \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, c - 1} x\right )\right )} \sqrt {\sqrt {c^{2} + 1} - 1}}{8 \, {\left (c^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{x^2}\,\sin \left (c\,x^2+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x^{2}} \sin {\left (a + c x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________